# Shear Force and Bending Moments

These tutorial questions consider statically determinate simply supported beams subjected to a variety of loading conditions. They get progressively more complex and are designed to be worked through sequentially. When selected, the span and loading conditions for each tutorial question are randomly generated giving an infinite number of possible questions to be attempted. Additionally there are randomly generated worked examples, printed tutorial sheets, study notes and video tutorials.

Each tutorial is selected by selecting the appropriate question number below. Notes:

• The analysis associated with these tutorial questions will round all your input values as follows:
• Forces, e.g. vertical and horizontal reactions (kN) - 1 decimal place.
• Bending moments (kNm) - 1 decimal place.
• Dimensions (m) - 3 decimal places.
• Numeric answers are required in all the fields. If you calculate a force or a moment as being zero, enter zero (0) in the appropriate field. If no answer is given it will be evaluated as zero.

• UDL - Universally Distributed Load; magnitude of load remains uniform throughout the length of the beam.
• PL - Point Load; concentrated load acting over a small distance, can be considered to be acting on a point.
• PUDL - Partial Universally Distributed Load; magnitude of load remains uniform throughout a portion length of the beam (not over the full span as with a UDL).
• TDL - Triangular Distributed Load; magnitude of load varies along the portion length of the beam over which it acts.
• VPDL - Varying Partially Distributed Load; made up of a partially distributed load (PDL) and a triangular distributed load (TDL), both acting over the same portion length of the beam. For the simply supported beam, calculate the following:

• The horizontal reactions.
• The vertical reactions.
• The position (X) from the support @ A, and the magnitude of the maximum bending moment.

Sketch the shear force and bending moment diagrams for the loaded the beam. Your sketches should indicate the values of the shear force at the critical locations and the magnitude/position of the maximum bending moment.

### Solution

@ A, HA=
kN
@ B, HB =
kN

@ A, RA =
kN
@ B, RB =
kN

#### Bending Moment

The position and magnitude of the bending moment from the support @ A.

X (from A)
m
@ X, MX
kNm For the simply supported beam, calculate the following:

• The horizontal reactions.
• The vertical reactions.
• The magnitude of the bending moment at the position (1) where the point load acts.

Sketch the shear force and bending moment diagrams and the deflected shape of the beam. Your sketches should indicate the values of the shear force at the critical locations and the magnitude/position of the maximum bending moment.

### Solution

@ A, HA=
kN
@ B, HB =
kN

@ A, RA =
kN
@ B, RB =
kN

#### Bending Moment

Magnitude of the bending moment at the position of the point load (@ 1).

@ 1, M1 =
kNm For the simply supported beam, calculate the following:

• The vertical reactions @ A & B.
• The magnitude and position of the maximum bending moment between A & B.
• The magnitude of the bending moment @ B.
• The position of the point of contraflexure (POC) between the supports @ A & B.

Sketch the shear force and bending moment diagrams and the deflected shape of the beam. Your sketches should indicate the values of the shear force at the critical locations and the magnitude/position of the maximum bending moment.

### Solution

@ A, RA =
kN
@ B, Rb =
kN

#### Bending Moment

The position and magnitude of the maximum bending moment in the main span between supports A & B.

X (from A)
m
@ X, MX =
kNm

The magnitude of the maximum bending moment at the support B.

@ B, MB
kNm

#### Point of Contraflexure between A & B

Calculate the position of the point of contraflexure between the supports A & B.

Note: Enter zero if there is no point of contraflexure.

XPOC (from A)
m For the simply supported beam, calculate the following:

• The vertical reactions @ A & B.
• The magnitude and position of the maximum bending moment between A & B.
• The magnitude of the bending moment at the points on the beam marked 1, 2, 3, 4 & C.
• The position of the point of contraflexure (POC) between the supports @ A & B.

Sketch the shear force and bending moment diagrams and the deflected shape of the beam. Your sketches should indicate the values of the shear force at the critical locations and the magnitude/position of the maximum bending moment.

### Solution

@ A, RA =
kN
@ B, RB =
kN

#### Bending Moment

The position and magnitude of the maximum bending moment in the main span between supports A & B.

If the beam between supports A & B is subject to a hogging moment i.e. tension in the top of the beam, then enter 0 for both X and MX.

X (from A)
m
@ X, MX =
kNm

The magnitude of the bending moment at the points on the beam marked 1, 2, 3, & 4.

@ 1, M1 =
kNm
@ 2, M2 =
kNm
@ 3, M3 =
kNm
@ 4, M4 =
kNm
@ The free end (C) =
kNm

#### Point of Contraflexure between A & B

Calculate the position of the point of contraflexure between the supports A & B.

Note: Enter zero if there is no point of contraflexure.

XPOC (from A)
m For the simply supported beam, calculate the following:

• The horizontal reaction at A.
• The vertical reactions at A & B.
• The bending moments at A & B, and the points marked 1, 2 & 3.
• The position and magnitude of the maximum bending moment between A & B.
• The position of the point(s) of contraflexure between A & B.

Sketch the shear force and bending moment diagrams and the deflected shape of the beam. Your sketches should indicate the values of the shear force at the critical locations and the magnitude/position of the maximum bending moment.

### Solution

@ A, HA =
kN

@ A, RA =
kN
@ B, RB =
kN

#### Bending Moment

The magnitude of the bending moment at A & B, and the points on the beam marked 1, 2, & 3.

@ A, MA =
kNm
@ B, MB =
kNm
@ 1, M1 =
kNm
@ 2, M2 =
kNm
@ 3, M3 =
kNm

The position (X) of the maximum positive bending moment measured from the support at A, and magnitude of the maximum bending moment @ X.

X (from A)
m
@ X, MX =
kNm

#### Point of Contraflexure between A & B

Calculate the position of the two points of contraflexure (X1 & X2) between the supports A & B.

Note: Enter zero for both X1 & X2 if there is no point of contraflexure, enter zero for X2 if there is only one point of contraflexure.

X1POC,1 (from A) =
m
X2POC,2 (from A) =
m For the simply supported beam, calculate the following:

• The vertical reactions @ A & B.
• The position and magnitude of the maximum bending moment.
• The magnitude of the bending moment @ points 1 & 2.

Sketch the shear force and bending moment diagrams and the deflected shape of the beam. Your sketches should indicate the values of the shear force at the critical locations and the magnitude/position of the maximum bending moment.

### Solution

@ A, RA =
kN
@ B, Rb =
kN

#### Bending Moment

The position (X) of the maximum positive bending moment measured from the support at A, and magnitude of the maximum bending moment @ X.

X (from A)
m
MEd @ X =
kNm

The magnitude of the bending moments at points 1 and 2.

@ 1, M1 =
kNm
@ 2, M2 =
kNm For the simply supported beam, calculate the following:

• The horizontal reaction at the support.
• The vertical reaction at the support.
• The magnitude of the bending moment at:
• The support.
• At a distance L2 from the support.
• At the position of the point load.
• At the free end.

Sketch the shear force and bending moment diagrams and the deflected shape of the beam. Your sketches should indicate the values of the shear force at the critical locations and the magnitude/position of the maximum bending moment. For the simply supported beam, calculate the following:

• The horizontal and vertical reactions @ A & B.
• The magnitude and position of the maximum bending moment between A & B.
• The magnitude of the bending moment at the points on the beam marked 1, 2, 3, & 4.
• The position of the point of contraflexure (POC) between the supports @ A & B.

Sketch the shear force and bending moment diagrams and the deflected shape of the beam. Your sketches should indicate the values of the shear force at the critical locations and the magnitude/position of the maximum bending moment.